Precalculus Examples

$y = 1800 {(\frac{1}{2})}^{\frac{x}{11}}$

Step 1

Write $y = 1800 {(\frac{1}{2})}^{\frac{x}{11}}$ as a function.

$f (x) = 1800 {(\frac{1}{2})}^{\frac{x}{11}}$

Step 2

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 3

Find the components of the definition.

Tap for more steps...

Step 3.1

Evaluate the function at $x = x + h$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 1800 {(\frac{1}{2})}^{\frac{x + h}{11}}$

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Simplify the expression.

Tap for more steps...

Step 3.1.2.1.1

Apply the product rule to $\frac{1}{2}$ .

$f (x + h) = 1800 (\frac{1^{\frac{x + h}{11}}}{2^{\frac{x + h}{11}}})$

Step 3.1.2.1.2

One to any power is one.

$f (x + h) = 1800 (\frac{1}{2^{\frac{x + h}{11}}})$

Step 3.1.2.2

Combine $1800$ and $\frac{1}{2^{\frac{x + h}{11}}}$ .

$f (x + h) = \frac{1800}{2^{\frac{x + h}{11}}}$

Step 3.1.2.3

The final answer is $\frac{1800}{2^{\frac{x + h}{11}}}$ .

$\frac{1800}{2^{\frac{x + h}{11}}}$

Step 3.2

Find the components of the definition.

$f (x + h) = \frac{1800}{2^{\frac{x + h}{11}}}$

$f (x) = \frac{1800}{2^{\frac{x}{11}}}$

$f (x + h) = \frac{1800}{2^{\frac{x + h}{11}}}$

$f (x) = \frac{1800}{2^{\frac{x}{11}}}$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{1800}{2^{\frac{x + h}{11}}} - (\frac{1800}{2^{\frac{x}{11}}})}{h}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

To write $\frac{1800}{2^{\frac{x + h}{11}}}$ as a fraction with a common denominator, multiply by $\frac{2^{\frac{x}{11}}}{2^{\frac{x}{11}}}$ .

$\frac{\frac{1800}{2^{\frac{x + h}{11}}} \cdot \frac{2^{\frac{x}{11}}}{2^{\frac{x}{11}}} - \frac{1800}{2^{\frac{x}{11}}}}{h}$

Step 5.1.2

To write $- \frac{1800}{2^{\frac{x}{11}}}$ as a fraction with a common denominator, multiply by $\frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}$ .

$\frac{\frac{1800}{2^{\frac{x + h}{11}}} \cdot \frac{2^{\frac{x}{11}}}{2^{\frac{x}{11}}} - \frac{1800}{2^{\frac{x}{11}}} \cdot \frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3

Write each expression with a common denominator of $2^{\frac{x + h}{11}} \cdot 2^{\frac{x}{11}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.1.3.1

Multiply $\frac{1800}{2^{\frac{x + h}{11}}}$ by $\frac{2^{\frac{x}{11}}}{2^{\frac{x}{11}}}$ .

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{x + h}{11}} \cdot 2^{\frac{x}{11}}} - \frac{1800}{2^{\frac{x}{11}}} \cdot \frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3.2

Multiply $2^{\frac{x + h}{11}}$ by $2^{\frac{x}{11}}$ by adding the exponents.

Tap for more steps...

Step 5.1.3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{x + h}{11} + \frac{x}{11}}} - \frac{1800}{2^{\frac{x}{11}}} \cdot \frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3.2.2

Combine the numerators over the common denominator.

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{x + h + x}{11}}} - \frac{1800}{2^{\frac{x}{11}}} \cdot \frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3.2.3

Add $x$ and $x$ .

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{2 x + h}{11}}} - \frac{1800}{2^{\frac{x}{11}}} \cdot \frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3.3

Multiply $\frac{1800}{2^{\frac{x}{11}}}$ by $\frac{2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{h}{11} + \frac{x}{11}}}$ .

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{2 x + h}{11}}} - \frac{1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{x}{11}} \cdot 2^{\frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3.4

Multiply $2^{\frac{x}{11}}$ by $2^{\frac{h}{11} + \frac{x}{11}}$ by adding the exponents.

Tap for more steps...

Step 5.1.3.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{2 x + h}{11}}} - \frac{1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{x}{11} + \frac{h}{11} + \frac{x}{11}}}}{h}$

Step 5.1.3.4.2

Combine the numerators over the common denominator.

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{2 x + h}{11}}} - \frac{1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{x + h + x}{11}}}}{h}$

Step 5.1.3.4.3

Add $x$ and $x$ .

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}}}{2^{\frac{2 x + h}{11}}} - \frac{1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.1.4

Combine the numerators over the common denominator.

$\frac{\frac{1800 \cdot 2^{\frac{x}{11}} - 1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.1.5

Simplify the numerator.

Tap for more steps...

Step 5.1.5.1

Factor $1800$ out of $1800 \cdot 2^{\frac{x}{11}} - 1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}$ .

Tap for more steps...

Step 5.1.5.1.1

Factor $1800$ out of $1800 \cdot 2^{\frac{x}{11}}$ .

$\frac{\frac{1800 (2^{\frac{x}{11}}) - 1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.1.5.1.2

Factor $1800$ out of $- 1800 \cdot 2^{\frac{h}{11} + \frac{x}{11}}$ .

$\frac{\frac{1800 (2^{\frac{x}{11}}) + 1800 (- 2^{\frac{h}{11} + \frac{x}{11}})}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.1.5.1.3

Factor $1800$ out of $1800 (2^{\frac{x}{11}}) + 1800 (- 2^{\frac{h}{11} + \frac{x}{11}})$ .

$\frac{\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{h}{11} + \frac{x}{11}})}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.1.5.2

Combine the numerators over the common denominator.

$\frac{\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{h + x}{11}})}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.1.5.3

Reorder $h$ and $x$ .

$\frac{\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{x + h}{11}})}{2^{\frac{2 x + h}{11}}}}{h}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{x + h}{11}})}{2^{\frac{2 x + h}{11}}} \cdot \frac{1}{h}$

Step 5.3

Combine.

$\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{x + h}{11}}) \cdot 1}{2^{\frac{2 x + h}{11}} h}$

Step 5.4

Simplify the expression.

Tap for more steps...

Step 5.4.1

Multiply $1800$ by $1$ .

$\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{x + h}{11}})}{2^{\frac{2 x + h}{11}} h}$

Step 5.4.2

Reorder factors in $\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{x + h}{11}})}{2^{\frac{2 x + h}{11}} h}$ .

$\frac{1800 (2^{\frac{x}{11}} - 2^{\frac{x + h}{11}})}{h \cdot 2^{\frac{2 x + h}{11}}}$

Step 6